Texture painting on sphere
I believe your problem in the screen shot you posted is because the direction of your normals don't match. Or it could be that you accidentally made two identical spheres overlapping. Without seeing your .blend I can't verify the nature of the problem. But there is a better way to texture a sphere anyway, and I will now share that with you.

1. Use a round cube instead

To get these menu options enable the Extra Objects mesh add-on in your preferences.



2. Set round cube's divisions as desired



3. UV Unwrap using Follow Active Quads



4. Scale to fit your UV layout and add an image texture



Rotate 180° if feeling unholy ;-)

(I'm just having a little fun, so I hope you guys won't get cross with me)